The Unapologetic Mathematician

Mathematics for the interested outsider

The Banach Space of Totally Finite Signed Measures

Today we consider what happens when we’re working over a \sigma-algebra — so the whole space X is measurable — and we restrict our attention to totally finite signed measures. These form a vector space, since the sum of two finite signed measures is again a finite signed measure, as is any scalar multiple (positive or negative) of a finite signed measure.

Now, it so happens that we can define a norm on this space. Indeed, taking the Jordan decomposition, we must have both \mu^+(X)<\infty and \mu^-(X)<\infty, and thus \lvert\mu\rvert(X)<\infty. We define \lVert\mu\rVert=\lvert\mu\rvert(X), and use this as our norm. It’s straightforward to verify that \lVert c\mu\rVert=\lvert c\rvert\lVert\mu\rVert, and that \lVert\mu\rVert=0 implies that \mu is the zero measure. The triangle inequality takes a bit more work. We take a Hahn decomposition X=A\uplus B for \mu+\nu and write

\displaystyle\begin{aligned}\lVert\mu+\nu\rVert&=\lvert\mu+\nu\rvert(X)\\&=(\mu+\nu)^+(X)+(\mu+\nu)^-(X)\\&=(\mu+\nu)(X\cap A)-(\mu+\nu)(X\cap B)\\&=\mu(A)+\nu(A)-\mu(B)-\nu(B)\\&\leq\lvert\mu\rvert(A)+\lvert\mu\rvert(B)+\lvert\nu\rvert(A)+\lvert\nu\rvert(B)\\&=\lvert\mu\rvert(X)+\lvert\nu\rvert(X)\\&=\lVert\mu\rVert+\lVert\nu\rVert\end{aligned}

So we know that this defines a norm on our space.

But is this space, as asserted, a Banach space? Well, let’s say that \{\mu_n\} is a Cauchy sequence of finite signed measures so that given any \epsilon>0 we have \lvert\mu_n-\mu_m\rvert(X)<\epsilon for all sufficiently large m and n. But this is larger than any \lvert\mu_n-\mu_m\rvert(E), which itself is greater than \lvert\lvert\mu_n\rvert(E)-\lvert\mu_m\rvert(E)\rvert. If E\subseteq A is a positive measurable set then this shows that \lvert\mu_n(E)-\mu_m(E)\rvert is kept small, and we find similar control over the measures of negative measurable sets. And so the sequence \{\mu_n(E)\} is always Cauchy, and hence convergent. It’s straightforward to show that the limiting set function \mu will be a signed measure, and that we will have control over \lVert\mu_n-\mu\rVert. And so the space of totally finite signed measures is indeed a Banach space.

June 28, 2010 Posted by | Analysis, Measure Theory | 2 Comments

   

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